FIN202 Topic 8: Pricing derivatives
Topic learning outcomes
On completing this topic, students should be able to: • calculate a futures contract price • explain different types of basis risk • construct a desired hedge • calculate the value of a physical bank bill and the value of a bank bill futures contract using the discount yield formula • calculate the value of three-year and 10-year bond futures contracts • analyse the six factors that influence the price of an option • construct basic pay-off diagrams as a visual representation of the risk-reward profile of options positions.
Interest rate changes
The dependence on interest rates is more difficult to determine. An increase in interest rates leads to an increase in the call option value and a decrease in the put option value.
Using beta factors for a more precise hedge
A beta factor is the measurement of the historical volatility of a particular stock or group of stocks relative to the market. By definition, the beta factor for the market as a whole is 1.0. If a stock has a high beta factor (more than 1.0), this means that in the past the stock has been, on average, more volatile than the market as a whole, while a stock with a beta factor of less than 1.0 has statistically shown less volatility than the market. If a certain stock has a beta factor of 1.10, for example, this conveys the information that in the past the stock has tended to move in the same direction as the whole market, however, the magnitude of this movement averaged about 10% more than the market's movement. It can be reasonably expected that the stock will exhibit similar behaviour relative to the market in the future, so that if the market has moved 5% in particular direction, the stock would be likely to move about 5.5% in the same direction.
Time left until options expiry
A longer time to expiry has the following effects on European options: • The call option premium will increase as the time value increases. • The time value of the put option will increase for larger values of the spot and decrease for smaller values of the spot (where the put is deep in-the-money) with longer time to expiry. For call options, an increase in the time to expiry has the same effect for both European and American call options — premiums increase. The effect of such an increase on American put options is different to that on European put options. It is connected with the fact that the extrinsic value of the American put is always greater than or equal to zero. For American puts, the value increases as the time to expiry increases.
Parallel moves in the yield curve
A parallel shift refers to a change in interest rates that is equivalent for all securities. Thus, the entire yield curve shifts up or down in parallel or by equivalent amounts at each maturity.
Anticipatory hedging
An anticipatory hedge involves taking a futures position as a substitute for a later cash transaction. An example is a wool merchant intending to buy wool in three months' time. They would hedge the risk of a rise in the wool price by fixing the purchase price through purchase of futures contracts today. An anticipatory hedger takes the same position toda
Volatility
An increase in volatility of the underlying asset will increase both the values of the call and the put. The volatility of the underlying asset also plays an important part in the valuation of an option. Volatility, in simple terms, is a measure of the degree to which an underlying asset's market price changes during a set period of time. If a commodity moves up or down to a greater degree, it is likely that the option over this underlying asset will be more expensive, reflecting this higher volatility or risk. The option should cost more to purchase, to compensate the writer for the greater risks involved in writing that option.
Option
An option is a contract between two parties in which one party (the buyer) has the right but not the obligation to buy or sell a specified asset at a specified price, at or before a specified date, from the other party (the seller). The seller of the option therefore has a contingent liability, or an obligation, which is activated if the buyer exercises that right. It is important to note that the buyer of the option is not obliged to complete the deal and will do so only if changes in price make it profitable to do so. The buyer of the option is protected from unfavourable market movements but is still able to profit from movements in the buyer's favour. The risk of loss is carried by the seller, who charges the buyer a fee for taking on this risk. This fee is called the premium. This feature of an option contract distinguishes it from other instruments, such as forward contracts or futures contracts, in which both parties to the contract have an obligation to transact at some time in the future. In an option contract, only one party (the seller) has an obligation to transact, but this is only if the other party (the buyer) requires them to do so. The act of enforcing the buyer's rights under an option contract is termed 'exercising the option'. As legally binding contracts, options create a legal relationship between the buyer and the seller of the option contract. This relationship can remain in place until the option is exercised or be allowed to lapse or expire (i.e. not exercised prior to, or on, the expiration date). Alternatively, it is possible for the two parties to the contract to enter into an opposite contract and for these contracts to be offset against each other.
Arbitraging with futures
As a futures contract allows for a future purchase or sale, the cash associated with this transaction has an opportunity cost. The difference between the futures price and the index price on any given day is in effect the difference between the interest earned on the cash and the dividend or income stream from the underlying asset, which is known as the cost of carry. When the difference is equal to the cost of carry, the stock and futures markets are said to be at their theoretical or fair value. Occasionally this relationship between prices temporarily breaks down, that is, the differential between the stock and the futures either widens or contracts, therefore, creating a special trading opportunity known as arbitrage. Arbitrageurs profit from taking advantage of these pricing anomalies across various markets or maturities within markets and their activity helps to return the pricing relationship back to fair value.
Amount hedged & timing
As many futures contracts are not settled in each month of the year, but only in the designated contract months, futures contracts are often entered into in a contract month in the future closest to, the time the cash transaction is made. This means cash and futures prices might not have yet converged to their greatest extent by the time the hedge is lifted, that is, when the futures contract is closed out and the final transaction is executed in the physical market. For example, 90-day bank-accepted bill futures are not settled in each month of the year — contract settlement months are March, June, September and December. Both lenders and borrowers, if they wish to use the bill futures market to hedge, should attempt to time borrowings and rollovers, so that they coincide with the futures maturity date to maximise the efficiency of the hedge. The hedge should be placed in a delivery month later than the timing of the cash transaction because to place it in an earlier month would force the futures contract to be closed out ahead of the physical purchase or sale, leaving the hedger exposed until the physical transaction occurred.
Australia's short-term interest rate market
Australia's short-term interest rate market consists of 90-day bank-accepted bill futures (bill futures) and 30-day interbank cash rate futures contracts. The 90-day bank-accepted bill future is one of the most actively traded contracts on the ASX and is the main focus of this topic. The 30-day interbank cash rate futures contract is based on the interbank overnight cash rate published by the Reserve Bank of Australia. It allows users to hedge against fluctuations in the overnight cash rate, and to better manage their daily cash exposures. The 30-day interbank cash rate future creates spread trading opportunities with existing ASX interest rate products, and arbitrage opportunities with over-the-counter products. Trading in bill futures is undertaken by many different types of market participants, including borrowers, lenders, market intermediaries and speculators. Bill futures provide these participants with a mechanism to manage some of the risks associated with fluctuations in interest rates. That is, bill futures are primarily used as a hedging instrument by borrowers, lenders and market intermediaries. Hedging in the interest rate futures market is exactly the same, in principle, as hedging in any other futures market. The arithmetic, however, is slightly more complicated because the contract values vary inversely with interest rates and are calculated according to a discount yield formula. These calculations, as well as specific techniques and strategies involved in trading bill futures, are explained throughout this topic.
Purpose of bond futures
Bond futures contracts are flexible and versatile financial tools that can be tailored to suit the requirements of the user. The following are typical uses of Commonwealth Treasury bond futures: • Holders of Commonwealth Government bonds and semi-government bonds, anticipating future sales from their investment portfolios, could sell Treasury bond futures to protect themselves against a rise in interest rates in the meantime. • Owners of Commonwealth Government and semi-government bonds wishing to protect their investment portfolios from the effect of interest rate rises, which would otherwise result in a fall in the value of the bonds, could sell bond futures contracts. • Bond dealers intending to participate in an expected bond tender in three months time could buy bond futures to protect themselves against a drop in interest rates in the meantime. (Commonwealth Government bonds are issued at periodic tenders.) • A financial institution, having bid unsuccessfully for stock at an official bond tender, but expecting bond rates to drop, could effectively take a position in bonds by buying bond futures. The contracts would then be liquidated when the expected drop in rates occurred, or when physical stock was purchased. • A bond dealer, having bought 10-year bonds at a tender in anticipation of a drop in interest rates, but experiencing difficulty finding a buyer when the drop in rates eventuates, could sell 10-year bond futures as an alternative to selling bonds. Sometime later, when a buyer for the stock has been found, the futures contracts would be liquidated; but, in the meantime, the profit on the transaction would be locked in. • A large superannuation fund, expecting maturing bonds to yield new investment funds in six months time, could buy bond futures to protect itself against the possibility of interest rates falling during that period. • An industrial company, intending to issue 10-year corporate bonds in several months time, could sell 10-year bond futures to offset any increases in borrowing costs caused by interest rate rises in the meantime. • Investment banks and short-term money market operators can use the contracts to facilitate bond switching, or trading the yield curve. A dealer anticipating a steepening of the yield curve and who wishes to sell 10-year bonds and buy five-year bonds to take advantage of this outlook, might not have 10-year bonds in their portfolio. Instead, they sell 10-year bond futures and buy five-year bonds in the physical market. Later, when the differential between five-year and 10-year bonds has increased, the dealer buys back 10-year bond futures and sells the five-year bonds to give an overall profit.
Commonwealth Treasury bond futures
Both the three-year and 10-year bond futures contracts are based on a hypothetical Commonwealth Treasury bond of $100,000 face value with a 6% coupon rate. The contract is cash-settled which means that on the last day of trading, all sold positions are automatically bought back and all bought positions are sold back at the cash settlement price. Unlike bill futures, it is not possible to take delivery of the underlying instrument.
Income stream
Certain assets provide a periodic income stream. For example, stocks pay a dividend, bonds pay a coupon and a foreign currency investment provides income in the form overseas interest. An increase in the level of income causes the growth rate of the asset to decrease and the potential payoff from the call option also decreases. The reverse holds true for a put option.
Exercise (strike) price
For calls, a larger strike price at a given spot price means that the intrinsic value of the call is less and so the premium will be less. The reverse is true for the put option.
Futures
Futures are contracts which are legally binding agreements, to buy or sell 'something' in the future. A futures contract may be established over a commodity such as gold, copper, wheat, corn, or a foreign exchange currency, share or share index or an interest rate financial instrument. Each contract specifies the quantity, quality, price and the date of delivery of the item. The buyer and seller of a futures contract agree on a price today for a product to be delivered and paid for on a predetermined date in the future. Futures are exchange traded. An over-the-counter futures contract is known as a forward. In the context of financial markets, the term 'option', essentially has the same meaning that it has in everyday life. The word relates to choice.
Hedging a current market position
Hedging a current market position involves taking a futures position that is opposite to the position the trader currently holds in the cash market. For example, an investment fund manager with a portfolio of shares could hedge against a fall in share prices, and hence a fall in the value of that portfolio, by selling SPI 200 futures contracts. The trader who wishes to hedge a current market position takes the opposite position in the futures market to that already held in the physical or cash market. Futures markets provide a way to offset (hedge) an exposure to price fluctuations in the physical market. Hedging, therefore, is about preserving wealth by reducing exposure to volatilities and price movements.
6 Factors influencing the price of an option
Ultimately, option prices are determined by supply and demand in the market. Six major factors influence the option price: • price of the underlying asset • exercise or strike price of the option • time left until the option's expiry • current interest rates • volatility of the underlying asset • any monetary return earned by holding the underlying asset (e.g. dividends in the case of equities).
Changes in the shape of the yield curve
Interest rates seldom move by the same magnitude all along the yield curve. For example, inflation news will tend to affect longer bonds more than shorter bonds, whereas a currency shock is more likely to affect the money market and shorter bonds. These non-uniform changes in interest rates affect the shape of the yield curve and could cause it to steepen, flatten or flex, depending on the relativities of the moves. A change in the shape of the yield curve is illustrated in Figure 2 below. Figure 2 Yield curve shift Buyers and sellers can determine appropriate rates and hence prices at which to transact by reference to a yield curve. Securities with different credit ratings are priced according to different yield curves, for example, lower rating securities are priced on a yield curve above the prime credit rates, that is rates for high rating securities (such as Government issued securities). Unlike shares, interest rate securities do not have a centralised exchange in most countries. That is, there is no central marketplace for buyers and sellers and no publicly observable price for the instruments. Each transaction is negotiated, often through an intermediary or designated broker screens. In Australia, bonds are mainly settled through two electronic clearing systems, RITS (Commonwealth Government bonds) and Austraclear (semi-government and some corporate securities). Similar settlement systems exist in other markets.
Hedging a current market position
Many institutions, such as life insurance companies, have large, highly diversified portfolios of shares that often closely match the performance of the market as a whole. Since the S&P/ASX 200 Index represents approximately 90% of the market value of all shares traded, ASX SPI 200 futures provide an excellent hedging vehicle for such portfolios, allowing managers to obtain protection against market falls. Faced with the possibility of a falling market, a portfolio manager can: • Liquidate the portfolio: This is expensive, given the brokerage costs associated with selling shares, the loss of entitlement to future dividends on the shares held and the downward effect on the cash market of selling a large portfolio. A capital gains tax liability might also be incurred when selling shares. • Sell futures against the portfolio: Any losses in the sharemarket will be approximately offset by gains in the futures market. The original shares remain in place throughout the expected market movement. Clearly, if the goal is to retain the portfolio through this period then selling futures is a much more efficient method of hedging against this risk. A portfolio manager who holds a large number of shares in various companies is subject to the risk that the market as a whole will fall. The manager is able to reduce that risk by selling ASX SPI 200 futures contracts against the value of the portfolio, which, in theory, can be used to virtually eliminate market risk from the portfolio. If the market should fall, the value of the portfolio would be protected (depending on the number of contracts sold), since losses would be offset by market gains. Similarly, if the market rises, there would be a loss on the futures market, but this would be offset by a corresponding gain in the value of the shares. In most cases, the loss or gain on futures will not exactly equal the fall or rise in corresponding portfolio value, but will give protection against a major portion of the market's expected fall in the sharemarket. Note: A third option is to use options to hedge against a fall in the value of the portfolio.
Basis risk
One of the advantages of futures is the convergence of futures and cash prices at expiry. However, cash and futures prices do not always exactly come together as the delivery month approaches. All hedgers should be aware that hedges against major price fluctuations may be imperfect and there is the risk that cash and futures price differences could slightly reduce the expected profit on the overall transaction. This risk is known as basis risk. Different types of basis risk are as follows: • Delivery basis: Basis risk usually relates to the cost of delivery under a deliverable futures contract, so it is often described as 'delivery basis'. The cost of delivery is related to the cost of funding, storing and insuring the commodity until delivery. Under cash-settled contracts, for which no delivery of a commodity occurs, this is not an issue as the cash settlement price and index or cash-price reference will be equal. • Grade basis: Hedging might not provide complete protection because the grade of the physical commodity in which the hedger trades might not match the standard futures grades. For example, in the wool example in section 2.1, the greasy wool futures contract is based on a grade of 21 micron. When using wool futures, there is a risk that a particular producer's output could vary from the 21 micron wool indicator of the futures contract. • Location basis: Prices tend to vary for markets in different areas. For example, a country wool grower using futures to hedge, but selling their wool nearby through private treaty, will need to take into account possible differences between the price they receive for their wool and the auction prices at the major selling centres. These variations are described as 'location basis' and are more significant for commodities such as agricultural products, than other types of futures.
Options trading - intrinsic versus time value
Option premiums (i.e. the price of an option) can be split into two separate components: • intrinsic value • extrinsic (time) value. An option's intrinsic value is based on the difference between its strike or exercise price (the price stated in the option description) and the current price of the underlying asset. For example, with gold at $754 per ounce, a December 725 gold call option would have $29 in intrinsic value (current price $754 - $725 strike). A call option has intrinsic value only if the asset price is above the strike price. A put option has an intrinsic value only if the asset price is below the strike price. Another way to approach intrinsic value is to imagine that the option is expiring today. If the option has a value, even with an immediate expiry, it has intrinsic value. Time value can best be described as the residual value of the option's premium above any intrinsic value. When an option is at-the-money (i.e. its trading price is at or near to the exercise price) or out-of-the-money (i.e. its trading price is below the exercise price in the case of a call, or above the exercise price in the case of a put), it has no intrinsic value and the entire premium (i.e. trading price of the option) represents time value. Time value declines as an option nears expiry, at which point it is zero (see Figure 4). Hence the longer the time until the expiration date, the higher an option's time value part of the premium. If the gold December 725 call option mentioned above had a premium of $35 and spot gold was trading at $754, its intrinsic value would be $29 and its time value $6 (or $35 premium less $29 intrinsic value).
Users of bond futures
Potential users of bond futures include any organisation that is exposed to changes in medium or long-term interest rates. Holders of Commonwealth Government bonds, dealers in bonds, corporate borrowers and lenders, and government authorities would find the futures market a useful vehicle through which to hedge against interest rate risks. Specifically, commercial banks, investment banks, money market dealers, insurance companies, finance companies, underwriting stockbrokers, superannuation funds, medium to large corporations and professional bond traders can all use bond futures to their advantage. Participants in the semi-government bond market are also large users of bond futures. These participants include the semi-government authorities themselves, and holders and dealers in semi-government bonds. Semi-government authorities act as public utilities for their respective state governments and their borrowings are usually guaranteed as to principal and interest by their governments. The main issuers in the semi-government bond market are: • New South Wales Treasury Corporation • Queensland Treasury Corporation • Western Australian Treasury Corporation • Treasury Corporation of Victoria • South Australian Financing Authority • Tasmanian Public Finance Corporation. The benchmark semi-government bond, the New South Wales Treasury Corporation (NSWTC) bond, is often priced directly against the three-year and 10-year bond futures contracts. Market practice is generally to price other semi-government bonds against the benchmark NSWTC issues, implying that bond futures are directly or indirectly relied upon to provide a base price for pricing the entire semi-government market. Spread trading between semi-government bonds and the bond futures contracts is a common strategy used to take advantage of perceived mispricing in semi-government bonds, relative to futures. These transactions can often affect bond futures prices throughout the course of the trading day.
Pay-off diagrams for the option seller
So far the topic has looked at a bought or long option position, which provides the buyer with limited risk and potentially unlimited reward. The opposite is true for the option seller (also known as the grantor or writer). The seller has limited profit potential (the most that can be earned is the option premium) and potentially unlimited risk.
International interest rate futures
The Australian market has a number of interest rate futures contracts. On exchanges in other countries, markets can include bond contracts based on both the government yield curve and other issuers or on securities backed by mortgages as part of a securitisation program. In the short-term interest rate market, several contracts on bank bills or money market paper of different maturities could be listed. The same or similar contracts could also be listed on more than one exchange. Two noticeable differences between the Australian bond contracts and most bond contracts listed elsewhere are as follows: • The bond contract is quoted as an index of 100 minus the yield (i.e. 98.24). Elsewhere, the actual contract value is quoted on a discount basis, normally as a cost per $100 face value (e.g. $98.24). • The Australian Securities Exchange (ASX) bond contracts are cash-settled whereas most other contracts have physical delivery contracts.
Price of the underlying asset
The price of the underlying asset, in relation to the exercise price of the option, is the single most important factor in determining the value of an option. If the price of an asset is either well above or well below the exercise price, other factors will have little influence on the option value. For calls, a larger spot price at a given exercise price will result in larger intrinsic value at expiry. This results in a larger premium for the call. An analogous argument shows that an increase in the spot price of the asset leads to a decrease in the put premium.
Who uses bank bill futures?
The primary purpose of futures is to manage risk exposure by protecting (hedging) against unfavourable market movements. Futures markets provide a mechanism for companies and institutions borrowing, lending or dealing in money or capital markets to separate the pricing of their transactions from the timing of the actual cash transactions. The careful use of futures enables a maximum or minimum price to be obtained before the transaction actually takes place. So who uses interest rate futures to hedge? These users can generally be divided into the following three main categories: • Borrowers in the marketplace might use interest rate futures as a hedging tool against adverse interest rate movements. Borrowers can include the following: - companies seeking funds for expansion and working capital - development and construction companies - institutions and semi-government bodies. • Lenders include companies and institutions that are suppliers of funds. Examples are insurance companies, superannuation funds and portfolio managers. • Market intermediaries include companies such as finance companies, together with institutions such as building societies. They also include those engaged in dealer-type activities, such as investment banks. This last group is oriented towards trading and may be involved in buying and selling large numbers of bank bills. They operate on narrow margins relative to the value of the bank bills and other securities in which they deal, and hence have a greater incentive to hedge their exposure to interest rate fluctuations than do most outright lenders or borrowers. Speculators also make use of interest rate futures. Speculators refers to individuals and companies that take positions in futures and/or options markets in anticipation of favourable moves in those markets to create profit. They are not hedgers. Speculators play an important role in any futures market as they create liquidity, and the bank bill futures market is no exception.
The yield
There are no comprehensive published rates for fixed interest securities, so holders of securities rely on information vendors to provide estimates of prices for the myriad of different tradeable securities. All fixed interest and pure discount securities in the Australian market trade at what is called the 'yield' or 'yield to maturity'. This is an indication of what would be earned on the bond if it was held to maturity with all the coupons reinvested at this same rate. There is a set formula that translates this yield into a price at which the transaction is conducted. Elsewhere in the world, bonds are quoted on price terms per $100 face value although short-term securities trade on a yield basis. The futures contracts follow the convention of the underlying market.
In-the-money/at-the-money/out-of-the-money
These terms are very important in the study of options and option trading and are defined as follows: • In-the-money: An option is in-the-money if it has intrinsic value. A December 725 gold call is in-the-money if the strike price of $725 is less than the market price of the gold. A December 725 gold put is in-the-money if the strike price is more than the market price of the gold. An option which has some intrinsic value must be in-the-money. • At-the-money: An option is at-the-money if the strike price of the option is the same as (or close to) the market price of the commodity. Hence, a December 725 call with the gold price at $725 would be an at-the-money call; similarly for the December 725 put. It has no intrinsic value. • Out-of-the-money: If the strike price of a call is more than the current market price of the commodity, or if the strike price of a put is less than the current market price of the commodity, the option is said to be out-of-the-money. Using the above gold example, a December 725 call is out-of-the-money if the current price of gold is below $725. It (again) has no intrinsic value. Note: An option where the strike price is very close, but not exactly the same as the current market price, is often still referred to as being at-the-money.
90-day bank-accepted bill futures
Trading in the 90-day bank-accepted bill futures contract at the ASX started in October 1979. The contract is based on a $1 million face value bank-accepted bill of exchange. The distinguishing feature of this contract is that it is a deliverable instrument. This means that if on the day of expiry of the futures contract a participant has a bought position they must receive delivery and if they have a sold position they must make delivery of physical bank bills.
Basics of interest rate futures -The yield curve
When thinking about interest rate futures, it is important to understand the time value of money (covered in Topic 5). Generally speaking, people are prepared to borrow money, and pay interest on it, because they prefer to have a certain quantity of goods now, rather than to save now and buy later. The interest rate is the price of using the money for a period of time. In general, short-term interest rates are lower than long-term interest rates, because the longer a person has funds invested at a certain rate in a market, the greater the likelihood that conditions in that particular market will change, meaning the lender is exposed to greater risk and therefore requires a higher interest rate. Groups of market securities with the same credit rating — for example, all Commonwealth Government bonds — display a relationship between their maturity date and the yield at which they trade. The chart of security yields versus their time to maturity is known as a yield curve. For a revision of yield curves, students can refer to Topic 5 of Financial Markets and Economic Principles (FIN101). The yield curve constantly moves in line with investor expectations. These moves are described as: • parallel moves in the yield curve • changes in the shape of the yield curve.
Basis in bond futures
Where the commodity being hedged is not a Commonwealth Government bond, e.g. if an industrial company is hedging a future issue of debentures, or an institution is hedging a portfolio of semi-government bonds (bonds issued by semi-government authorities), this difference in credit quality means that the futures contract and the security being hedged might not move by exactly the same amounts. This introduces an element of basis risk into the hedge (see Topic 3).
Is it possible to make a gain if the ASX SPI 200 futures price is higher than the theoretical fair price given by the cost of holding shares? If so, describe the strategy an investor could use in this situation.
Yes, it is possible to make a gain if the ASX SPI 200 futures price is higher than the theoretical fair price. In this case an arbitrageur would buy the physical shares and simultaneously sell ASX SPI 200 futures. When the futures contract matures, the shares would be sold and the futures contracts closed out.